Local linear regression for functional data under quasi-associated dependence with fixed and $ k $NN bandwidths

Wahiba Bouabsa; Sadiah M. Aljeddani; Wahiba Bouabsa; Sadiah M. Aljeddani

doi:10.3934/math.2026641

AIMS Mathematics

2026, Volume 11, Issue 6: 15581-15625. doi: 10.3934/math.2026641

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Local linear regression for functional data under quasi-associated dependence with fixed and $ k $NN bandwidths

Wahiba Bouabsa ^{1
,
,},
Sadiah M. Aljeddani ²

1.
Laboratory of Statistics and Stochastic Processes, University of Djillali Liabes BP 89, Sidi Bel Abbes 22000, Algeria
2.
Department of Mathematics, Al-Lith University College, Al Lith 21961, Umm Al-Qura University, Makkah, Saudi Arabia

Received: 23 January 2026 Revised: 07 April 2026 Accepted: 28 April 2026 Published: 03 June 2026
MSC : 62G07, 62G20, 62G35, 62H12

Local linear kernel estimation is a fundamental tool in nonparametric regression, renowned for its bias reduction and boundary correction properties. While its asymptotic behavior is well understood for independent data and for strongly mixing processes, it remains largely unexplored under quasi-associated dependence, even in the real-valued regression setting. In this paper, we introduce a functional local linear kernel estimator for regression models with quasi-associated observations. We establish strong consistency in the sense of almost complete convergence and derive convergence rates under mild regularity conditions involving small-ball probabilities and covariance decay. To the best of our knowledge, this paper provides the first theoretical guarantees for functional local linear kernel regression under quasi-associated dependence, covering both fixed and $ k $-nearest neighbor ($ k $NN) bandwidth selection procedures.
- functional data,
- local linear estimation,
- quasi-association,
- almost complete convergence,
- kernel methods,
- $ k $NN method
Citation: Wahiba Bouabsa, Sadiah M. Aljeddani. Local linear regression for functional data under quasi-associated dependence with fixed and $ k $NN bandwidths[J]. AIMS Mathematics, 2026, 11(6): 15581-15625. doi: 10.3934/math.2026641

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Abstract

Local linear kernel estimation is a fundamental tool in nonparametric regression, renowned for its bias reduction and boundary correction properties. While its asymptotic behavior is well understood for independent data and for strongly mixing processes, it remains largely unexplored under quasi-associated dependence, even in the real-valued regression setting. In this paper, we introduce a functional local linear kernel estimator for regression models with quasi-associated observations. We establish strong consistency in the sense of almost complete convergence and derive convergence rates under mild regularity conditions involving small-ball probabilities and covariance decay. To the best of our knowledge, this paper provides the first theoretical guarantees for functional local linear kernel regression under quasi-associated dependence, covering both fixed and $ k $-nearest neighbor ($ k $NN) bandwidth selection procedures.

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